In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set ${\displaystyle \operatorname {C} _{G}(S)}$ of elements of G that commute with every element of S, or equivalently, the set of elements ${\displaystyle g\in G}$ such that conjugation by ${\displaystyle g}$ leaves each element of S fixed. The normalizer of S in G is the set of elements ${\displaystyle \mathrm {N} _{G}(S)}$ of G that satisfy the weaker condition of leaving the set ${\displaystyle S\subseteq G}$ fixed under conjugation. The centralizer and normalizer of S are subgroups of G. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets S.

Suitably formulated, the definitions also apply to semigroups.

In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra.

The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

The centralizer of a subset ${\displaystyle S}$ of group (or semigroup) G is defined as

where only the first definition applies to semigroups. If there is no ambiguity about the group in question, the G can be suppressed from the notation. When ${\displaystyle S=\{a\}}$ is a singleton set, we write C_G(a) instead of C_G({a}). Another less common notation for the centralizer is Z(a), which parallels the notation for the center. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, Z(g).

The normalizer of S in the group (or semigroup) G is defined as

where again only the first definition applies to semigroups. If the set ${\displaystyle S}$ is a subgroup of ${\displaystyle G}$, then the normalizer ${\displaystyle N_{G}(S)}$ is the largest subgroup ${\displaystyle G'\subseteq G}$ where ${\displaystyle S}$ is a normal subgroup of ${\displaystyle G'}$. The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of ${\displaystyle S}$ and s is in ${\displaystyle S}$, then it must be that gs = sg, but if g is in the normalizer, then gs = tg for some t in ${\displaystyle S}$, with t possibly different from s. That is, elements of the centralizer of ${\displaystyle S}$ must commute pointwise with ${\displaystyle S}$, but elements of the normalizer of S need only commute with S as a set. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the normal closure.